1
课程详述
COURSE SPECIFICATION
以下课程信息可能根据实际授课需要或在课程检讨之后产生变动。如对课程有任何疑问，请联
系授课教师。
The course information as follows may be subject to change, either during the session because of unforeseen
circumstances, or following review of the course at the end of the session. Queries about the course should be
directed to the course instructor.
1.
课程名称 Course Title
随机分析 Stochastic Analysis
2.
授课院系
Originating Department
数学系 Mathematics
3.
课程编号
Course Code
MAT7029
4.
课程学分 Credit Value
3
5.
课程类别
Course Type
专业选修课 Major Elective Courses
6.
授课学期
Semester
春季 Spring
7.
授课语言
Teaching Language
中英双语 English & Chinese
8.
授课教师、所属学系、联系方
式（如属团队授课，请列明其
他授课教师）
Instructor(s), Affiliation&
Contact
（For team teaching, please list
all instructors）
熊捷，讲座教授，数学系
慧园 3 栋 527
Jie Xiong, Chair Professor, Department of Mathematics
Block 3 Room.527, Wisdom Valley
9.
实验员/助教、所属学系、联系
方式
Tutor/TA(s), Contact
待公布 To be announced
10.
选课人数限额(可不填)
Maximum Enrolment
（Optional）
授课方式
Delivery Method
习题/辅导/讨论
Tutorials
实验/实习
Lab/Practical
其它(请具体注明)
Other（Please specify）
总学时
Total
11.
学时数
Credit Hours
48
2
12.
先修课程、其它学习要求
Pre-requisites or Other
Academic Requirements
MA215 Probability Theory， MA301 Theory of Functions of a Real
Variable；MA215 概率论，MA301 实变函数
13.
后续课程、其它学习规划
Courses for which this course
is a pre-requisite
14.
其它要求修读本课程的学系
Cross-listing Dept.
教学大纲及教学日历 SYLLABUS
15.
教学目标 Course Objectives
这门课的目标是在概率论的基础上，掌握随机分析的基础理论与方法，并会介绍基本的随机微分方程理论和倒
向随机微分方程理论。
This course, based on the preliminary knowledge of Probability Theory, will discuss the basic theory and
methods in Stochastic Analysis. It then introduces Stochastic Differential Equations Theory. The basic theory
and methods of Backward Stochastic Differential Equations will also be covered.
16.
预达学习成果 Learning Outcomes
1.掌握随机分析的基础理论与方法。
2.了解基础的随机微分方程理论。
3.了解基本的倒向随机微分方程的理论与方法。
1. Master the basic theory and methods of Stochastic Analysis.
2. Understand the basic theory of. Stochastic Differential Equations.
3. Understand the basic theory and methods of Backward Stochastic Differential Equations.
17.
课程内容及教学日历 （如授课语言以英文为主，则课程内容介绍可以用英文；如团队教学或模块教学，教学日历须注明
主讲人）
Course Contents (in Parts/Chapters/Sections/Weeks. Please notify name of instructor for course section(s), if
this is a team teaching or module course.)
3
1. 初步（8 课时）：条件期望；离散时间鞅; Doob 可选抽样定理; 上、下鞅; 鞅不等式；鞅收敛定理；连续时
间鞅等。
Preliminaries (8 hours): Conditional Expectation; Discrete time Martingale; Doob’s Optional Sampling
Theorems; Supermartingales and Submartingales; Martingale Inequalities; Martingale Convergence
Theorems, Introduction to Continuous Time Martingales, etc.
2. 布朗运动（8 课时）：布朗运动定义、分布，流；鞅性质；二次变差；马尔科夫性；首中时；反射原理
等。
Brownian Motion(BM) (8 hours): Definition of BM; Distribution of BM; Filtration for BM; Martingale
Property for BM; Quadratic Variation; Markov Property of BM; First Passage Time Distributions;
Reflection Principle, etc.
3. Itô 积分（16 课时）：Itô 积分定义；Itô-Doeblin 公式;分部积分；随机 Fubini 定理；Girsanov 定理；布朗
鞅与表示定理；Feynman-Kac 表示等。
Itô Integrals (16 hours): Definition of Itô’s Integral; Itô-Doeblin Formula; Integration by parts; Stochastic
Fubini Theorem; The Girsanov Theorem; The Brownian Martingale Representation Theorem; The
Feynman-Kac Representation, etc.
4. 随机微分方程理论（8 课时）
Stochastic Differential Equations (8 hours)
5. 倒向随机微分方程（8 课时）
Backward Stochastic Differential Equations (8 hours)
18.
教材及其它参考资料 Textbook and Supplementary Readings
1. J. Xiong. An Introduction to Stochastic Filtering Theory. Oxford Graduate Texts in Mathematics,
18. Oxford University Press, 2008.
2. Jiongmin Yong and Xunyu Zhou, Stochastic Control: Hamiltonian Systems and HJB Equations
课程评估 ASSESSMENT
19.
评估形式
Type of
Assessment
评估时间
Time
占考试总成绩百分比
% of final
score
违纪处罚
Penalty
备注
Notes
出勤 Attendance
课堂表现
Class
Performance
小测验
Quiz
课程项目 Projects
平时作业
Assignments
20%
期中考试
Mid-Term Test
30%
期末考试
Final Exam
50%
期末报告
Final
Presentation
4
其它（可根据需要
改写以上评估方
式）
Others (The
above may be
modified as
necessary)
20.
记分方式 GRADING SYSTEM
A. 十三级等级制 Letter Grading
B. 二级记分制（通过/不通过） Pass/Fail Grading
课程审批 REVIEW AND APPROVAL
21.
本课程设置已经过以下责任人/委员会审议通过
This Course has been approved by the following person or committee of authority
数学系课程规划与审核委员会
Curriculum Planning and Review Committee, Department of Mathematics